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# If all of the telephone extensions in a certain company must

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Re: If all of the telephone extensions in a certain company must be even [#permalink]
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Bunuel wrote:
If all of the telephone extensions in a certain company must be even numbers, and if each of the extensions uses all four of the digits 1, 2, 3, and 6, what is the greatest number of four-digit extensions that the company can have?

(A) 4
(B) 6
(C) 12
(D) 16
(E) 24

Take the task of arranging the 4 digits and break it into stages.

We’ll begin with the most restrictive stage.

Stage 1: Select the digit in the units position
Since the 4-digit extension must be EVEN, the unit digit must be either 2 or 6
So, we can complete stage 1 in 2 ways

Stage 2: Select the tens digit
There are 3 remaining digits from which to choose, so we can complete this stage in 3 ways.

Stage 3: Select the hundreds digit
There are 2 digits remaining, so we can complete this stage in 2 ways.

Stage 4: Select the thousands digit
There 1 digit remaining, so we can complete this stage in 1 way.

By the Fundamental Counting Principle (FCP), we can complete all 4 stages (and thus create an even extension) in (2)(3)(2)(1) ways (= 12 ways)

NOTE: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. For more information about the FCP, watch our free video: https://www.gmatprepnow.com/module/gmat-counting/video/775

Cheers,
Brent
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If all of the telephone extensions in a certain company must be even [#permalink]
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Kudos
Bunuel wrote:
If all of the telephone extensions in a certain company must be even numbers, and if each of the extensions uses all four of the digits 1, 2, 3, and 6, what is the greatest number of four-digit extensions that the company can have?

(A) 4
(B) 6
(C) 12
(D) 16
(E) 24

NOTE: Always SCAN the answer choices before performing any calculations.
Here, we notice that the answer choices are pretty small, so if we don't come up a full solution, we should consider LISTING AND COUNTING

We get the following extensions:
1236
1326
1362
1632
2136
2316
3126
3162
3216
3612
6132
6312
TOTAL = 12

Cheers,
Brent

Originally posted by BrentGMATPrepNow on 23 Nov 2015, 12:57.
Last edited by BrentGMATPrepNow on 10 Aug 2020, 05:47, edited 1 time in total.
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Re: If all of the telephone extensions in a certain company must [#permalink]
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Dr_Friedrich wrote:
If all of the telephone extensions in a certain company must be even numbers, and if each of the extensions uses all four of the digits 1, 2, 3, and 6, what is the greatest number of four-digit extensions that the company can have?

(A) 4
(B) 6
(C) 12
(D) 16
(E) 24

Solution:

If we let 6 be the last digit, then we have 3! = 6 options for the other 3 digits. If we let 2 be the last digit, we have 6 options also. Therefore, we have a total of 12 options.

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Re: If all of the telephone extensions in a certain company must be even [#permalink]
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Bunuel wrote:
If all of the telephone extensions in a certain company must be even numbers, and if each of the extensions uses all four of the digits 1, 2, 3, and 6, what is the greatest number of four-digit extensions that the company can have?

(A) 4
(B) 6
(C) 12
(D) 16
(E) 24

If we let 6 be the last digit, then we have 3! = 6 options for the other 3 digits. If we let 2 be the last digit, we have 6 options also. Therefore, we have a total of 12 options.

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Re: If all of the telephone extensions in a certain company must [#permalink]
Hello,
This is probably a silly question...but why 4 cannot be in the unit place? isn't' 4 also an even number?
Thank you!
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Re: If all of the telephone extensions in a certain company must [#permalink]
YinLZ wrote:
Hello,
This is probably a silly question...but why 4 cannot be in the unit place? isn't' 4 also an even number?
Thank you!

We are told that only 1, 2, 3, and 6 for the extensions, so as you can see 4 is not among the numbers allowed.
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Re: If all of the telephone extensions in a certain company must [#permalink]
BrentGMATPrepNow wrote:
Bunuel wrote:
If all of the telephone extensions in a certain company must be even numbers, and if each of the extensions uses all four of the digits 1, 2, 3, and 6, what is the greatest number of four-digit extensions that the company can have?

(A) 4
(B) 6
(C) 12
(D) 16
(E) 24

Take the task of arranging the 4 digits and break it into stages.

We’ll begin with the most restrictive stage.

Stage 1: Select the digit in the units position
Since the 4-digit extension must be EVEN, the unit digit must be either 2 or 6
So, we can complete stage 1 in 2 ways

Stage 2: Select the tens digit
There are 3 remaining digits from which to choose, so we can complete this stage in 3 ways.

Stage 3: Select the hundreds digit
There are 2 digits remaining, so we can complete this stage in 2 ways.

Stage 4: Select the thousands digit
There 1 digit remaining, so we can complete this stage in 1 way.

By the Fundamental Counting Principle (FCP), we can complete all 4 stages (and thus create an even extension) in (2)(3)(2)(1) ways (= 12 ways)

NOTE: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. For more information about the FCP, watch our free video: https://www.gmatprepnow.com/module/gmat-counting/video/775

Cheers,
Brent

Hi BrentGMATPrepNow, Question asked what is the greatest number of four-digit extensions that the company can have? Which mean numbers doesn't need to be different and digit can repeat except the last digit of either 2 or 6. Therefore not sure why is not 4 * 4 * 4 * 2 = 128 ?
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Re: If all of the telephone extensions in a certain company must [#permalink]
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Kimberly77 wrote:
Hi BrentGMATPrepNow, Question asked what is the greatest number of four-digit extensions that the company can have? Which mean numbers doesn't need to be different and digit can repeat except the last digit of either 2 or 6. Therefore not sure why is not 4 * 4 * 4 * 2 = 128 ?

The question tells us "...each of the extensions uses all four of the digits 1, 2, 3, and 6"
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Re: If all of the telephone extensions in a certain company must [#permalink]
BrentGMATPrepNow wrote:
Kimberly77 wrote:
Hi BrentGMATPrepNow, Question asked what is the greatest number of four-digit extensions that the company can have? Which mean numbers doesn't need to be different and digit can repeat except the last digit of either 2 or 6. Therefore not sure why is not 4 * 4 * 4 * 2 = 128 ?

The question tells us "...each of the extensions uses all four of the digits 1, 2, 3, and 6"

Thanks BrentGMATPrepNow. all four here mean it's unique/different. Noted thanks Brent.
In that case, could we presume that when it comes to numbers like telephone extension, codes...etc which position matters. it's presumed to be different?
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Re: If all of the telephone extensions in a certain company must [#permalink]
Top Contributor
Kimberly77 wrote:
BrentGMATPrepNow wrote:
Kimberly77 wrote:
Hi BrentGMATPrepNow, Question asked what is the greatest number of four-digit extensions that the company can have? Which mean numbers doesn't need to be different and digit can repeat except the last digit of either 2 or 6. Therefore not sure why is not 4 * 4 * 4 * 2 = 128 ?

The question tells us "...each of the extensions uses all four of the digits 1, 2, 3, and 6"

Thanks BrentGMATPrepNow. all four here mean it's unique/different. Noted thanks Brent.
In that case, could we presume that when it comes to numbers like telephone extension, codes...etc which position matters. it's presumed to be different?

I wouldn't say that all four implies that the words must be different.
In general, if a question asks you to determine the total number of outcomes, I would say that it's implied that each outcome is different.
For example, for this question we wouldn't count the extensions 1263 and 1263 as different outcomes since they are the same numbers.

I would say that in the directions "...each of the extensions uses all four of the digits 1, 2, 3, and 6", all four means:
The digit 1 must appear in the extension.
And, the digit 2 must appear in the extension.
And, the digit 3 must appear in the extension.
And, the digit 6 must appear in the extension.
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Re: If all of the telephone extensions in a certain company must [#permalink]
Noted. Thanks BrentGMATPrepNow
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Re: If all of the telephone extensions in a certain company must [#permalink]
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